Vibration data pre-processing techniques for rolling element bearing ...

23rd International Congress on Sound & Vibration Athens,

Greece

10-14 July 2016

ICSV23

VIBRATION DATA PRE-PROCESSING TECHNIQUES FOR ROLLING ELEMENT BEARING FAULT DETECTION Cédric Peeters, Patrick Guillaume and Jan Helsen University of Brussels – VUB, Faculty of Mechanical Engineering, 1050 Elsene, Belgium email: [email protected] Real world vibration measurements often contain a lot of signal components originating from different machine elements. Detecting and recognizing the signal of a single element is thus a complicated endeavor because of the masking presence of other element’s signals in the measured data. Of particular interest to the industry is the early detection of faults in rolling element bearings. Normally the signal strength of a faulty bearing is however weaker than the ones of more prominent signals coming from other mechanical elements like gearboxes, rotating shafts and/or mechanical loads. These stronger deterministic signals form discrete components in the frequency domain. A number of techniques have been developed to filter out these components in order to detect the fault signals of the bearing elements more easily, e.g. time-synchronous averaging (TSA), self-adaptive noise cancellation (SANC), discrete/random separation (DRS) and cepstral editing procedure (CEP). The latter technique has proven to be particularly promising in filtering out deterministic harmonic content in the measured signal while enhancing the bearing fault signals that can be treated as cyclostationary. If the shaft speed variation is limited, the cepstral editing procedure can remove selected discrete frequency components without order tracking. In the cepstral domain the narrow impulses from the deterministic components can be removed or liftered by applying a notch lifter on the cepstrum. Since manually choosing which parts to lifter is arduous, an automated CEP method (ACEP) is developed that detects the cepstral peaks without user interaction and that tries to outperform the traditional cepstral editing procedure. In a general condition monitoring scheme, going from measured signal to fault detection & diagnosis, this ACEP method is an interesting precursor to other analysis techniques like spectral kurtosis and envelope analysis.

1.

Introduction

Rolling element bearings are one of the most widely used machine elements in rotating machinery. It is estimated that more than 90% of rotating machines [1] contain this type of bearings. Unfortunately, they are susceptible to a multitude of premature deficiencies. Less than 10% of rolling element bearings [2] reach their expected basic L10 life, the life at which ten percent of the bearings can be expected to have failed due to normal fatigue failure for that particular application. These observations imply a need for an improved comprehensive condition-based maintenance program. However there are still some hurdles to be overcome before such an exhaustive program becomes fully feasible. One of these hurdles is the detection of fundamental bearing fault frequencies that are masked by high energy harmonic signals originating from other machine elements like shafts or gears. This paper focuses on separating the bearing fault signals from other masking signal content of such elements. The separation is based on the assumption that bearing fault signals are stochastic due to random jitter on their fundamental period and can be categorized as cyclostationary whereas gear or shaft signals are deterministic and appear as distinct peaks in the amplitude spectrum [3]. 1

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Recently there have been a number of developments concerning the topic of discrete components removal (DCR). Examples of such preprocessing techniques include time-synchronous averaging (TSA) [4], self-adaptive noise cancellation (SANC) [5], linear prediction filtering (LPF) [6], discrete/random separation (DRS) [7] and cepstral editing (CEP) [8]. This last preprocessing technique is investigated further in this paper as it shows great potential in separating cyclostationary bearing signals from discrete components [9]. Qualitative studies by Kilundu et al. [10] and Randall et al. [8] have been conducted comparing cepstral editing to the other aforementioned methods. These studies indicated through experimental tests that the CEP technique can outperform the other methods. The aim of this paper is to demonstrate the performance of the cepstral editing procedure by applying it to simulated vibration signals. Firstly, some background information about the cepstrum is presented, followed by an explanation of the cepstrum editing procedure (CEP) and a description of the necessary changes for an automated scheme. In the second section the performance of the automated cepstrum editing procedure (ACEP) will be assessed by investigating the residual signals after ACEP on virtual data sets.

2.

The cepstrum

2.1

Background

Cepstrum analysis is a signal processing tool that has been available for quite a while now. The history of the cepstrum goes back to 1963 when it was first introduced by Bogert et al. [11]. The definition used then as "the power spectrum of the logarithmic power spectrum" has since evolved and other definitions for the cepstrum have been distinguished afterward, like the "complex cepstrum" by Oppenheim and Schafer [12]. Various discussions were held about the proper definitions and properties of the cepstrum [13, 14] and different forms were described like the differential cepstrum [15] and the mean differential cepstrum [16], which have their use mainly in operational modal analysis (OMA). Until recently however the cepstrum was not used that often in the field of vibration-based condition monitoring, its applications were primarily related to speech analysis and echo detection in seismic signals[17]. In the last few years, the interest in cepstrum analysis for condition monitoring purposes has increased significantly. It is now understood that the real cepstrum can be used to edit the log amplitude spectrum of stationary signals and combined with the original phase to achieve edited time signals. This last finding is of particular use for pre-processing vibration data originating from accelerometers and is the main topic of this paper. 2.2

Formulations

The complex cepstrum is defined as the inverse Fourier transform of the log spectrum. It is here expressed in terms of the amplitude and the phase of the spectrum: Cc (τ ) = F −1 {log(X(f ))} = F −1 {ln(A(f )) + jφ(f )}

(1)

where X(f ) is the frequency spectrum of the signal x(t): X(f ) = F {x(t)} = A(f )ejφ(f )

(2)

By setting the phase to zero in Eq.(1), the real cepstrum can be obtained: Cr (τ ) = F −1 {ln(A(f ))}

(3)

Here, τ is a measure of time, referred to as "quefrency", however it is not defined in the same sense as a signal in the time domain. A peak at a certain quefrency corresponds to the inverse period of a series of periodic harmonics in the spectrum. For example, if the sampling rate of a signal is 20 kHz 2

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and the cepstrum displays a quefrency peak at 1000 samples, the peak indicates that there is a family of harmonics present in the spectrum with a spacing of 20 Hz (20 kHz/1000 samples). An important property of the cepstral domain is that the convolution of two time domain signals can be expressed as an addition of their cepstra. Suppose an output signal y(t) of a physical system that is the convolution of an input signal x(t) and an impulse response h(t) of the system: y(t) = x(t) ∗ h(t)

(4)

Because of the convolution theorem, this time domain expression transforms into a multiplication in the frequency domain: Y (f ) = X(f )H(f ) (5) In turn, taking the logarithm of Eq.(5) transforms the multiplication into a sum: log(Y (f )) = log(X(f )) + log(H(f ))

(6)

Since the Fourier transform is a linear transform, the addition remains valid in the cepstral domain. C(τ ) = F −1 {log(Y (f ))} = F −1 {log(X(f ))} + F −1 {log(H(f ))}

(7)

This property indicates the possibility to deconvolve a signal if one of the factors is known. As such the logarithmic transformation allows to separate the influence of the excitation source and the transmission path of the system in the cepstral domain. 2.3

Cepstrum editing procedure

The cepstrum, being the inverse Fourier transform of a log spectrum, is able to concentrate periodic spectrum components, e.g. harmonic families and sidebands into a smaller number of narrow impulses called "rahmonics". This property creates opportunities for editing these peaks in the cepstral domain (also called "liftering") as to decrease the amount of harmonic signal components present in the signal. Since the complex cepstrum contains both amplitude and phase information, it is possible to reverse the cepstral signal back to the time domain after editing. However, to be able to calculate the complex cepstrum through Eq.(1), the phase information φ(f ) needs to be unwrapped to a continuous frequency function. This complicates the use of the complex cepstrum with stationary signals consisting of discrete frequency components, where the phase is undefined at intermediate frequencies, and of stationary random components for which the phase varies randomly for every frequency. Recently though, Randall and Sawalhi [18] showed that the real cepstrum can be used for editing the amplitude spectrum of signals with harmonic content. The edited real cepstrum can be transformed back to the amplitude spectrum and this can then be recombined with the original phase as to return to the time domain. This cepstrum editing procedure (CEP) has been proposed to be used as a means of removing harmonic frequencies from signals and thus separating the deterministic signal content from the random or cyclostationary signal content. The CEP method has been investigated further by Randall et al. [8] and a general scheme of the procedure is shown in Fig.1. Due to the editing of the amplitude at certain frequencies, these frequencies will also exhibit an incorrect phase. However, the amplitude reduction of the discrete components is significant, generally to the level of neighboring noise components, while the phase distortion will mostly be negligible [18]. In Fig.1 it is seen that the most crucial step is the editing of the real cepstrum. While Randall and Sawalhi [8] choose for a manual editing approach based on knowledge about the examined machine configuration, there have recently been developments to design an automated editing procedure [9], which is a more desirable approach for industrial applications. In this paper the concept of an automated cepstrum editing procedure is expounded. ICSV23, Athens (Greece), 10-14 July 2016

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Figure 1: Schematic diagram of the cepstrum editing procedure. 2.4

Automated cepstral editing

Replacing the manual procedure by an automated one entails the need for added intelligence in selecting the proper cepstral peaks. Before this selection can be made though, the cepstrum of the signal requires enhancement as to make the peak detection step as straightforward as possible. Enhancing of the cepstrum consists of two main steps. First, a long-pass lifter (equivalent to a highpass filter in the frequency domain) is applied to the cepstrum in order to prevent the liftering of low quefrency content. Second, the presence of noise in the signal is reduced through the use of wavelet denoising and the spectral subtraction (SS) method. After the cepstrum has been sufficiently enhanced, the automatic peak detector generates liftering intervals corresponding to the cepstral peaks resulting essentially in a notch lifter. Finally, the cepstrum is multiplied with this notch lifter to produce the edited real cepstrum. An overview of the fully automated procedure can be seen on Fig.2.

Figure 2: Schematic diagram of the automated cepstrum editing step.

2.4.1

Long-pass lifter

Before performing peak detection and removal on the real cepstrum, it is necessary to remove the low quefrency content before enhancement of the cepstrum. This implies that there will be no detected peaks in the low quefrency region and correspondingly there will be no liftering of low quefrency content on the original real cepstrum. A property of the cepstral transformation is that it relegates modal information to the low quefrency region and because it is desirable to preserve the system resonances for the purpose of bandpass filtering in envelope analysis, this region is left untouched. While it is possible to base the decision of the cut-off quefrency on the resonance frequencies present in the system under investigation [9], properly determining the cut-off quefrency is difficult for a general case and involves manual inspection of the signal content. However in order to automate the process, it is reasonable to define a long-pass lifter with a high cut-off quefrency (>0.005s) in the interest of having a safety margin to prevent liftering away too much modal content. 4

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The long-pass lifter can be defined as:  0 lLP (n) = 1

n = 1 : Ncut−of f n = Ncut−of f + 1 : L

(8)

where n, Ncut−of f and L are respectively the sample quefrency index, the cut-off quefrency index and the sample length of the cepstrum. Because of the symmetry imposed by the discrete Fourier transform, the real cepstrum can just be analyzed until half of the signal length, i.e. L = bN/2c. If c(n) represents the unedited real cepstrum, the long-pass liftered cepstrum can be calculated as follows: cLP (n) = c(n)lLP (n). (9) 2.4.2

Noise reduction

In order to detect cepstral peaks more efficiently, the cepstrum is denoised using two conventional denoising methods. As proposed in [9], the first applied method is spectral subtraction (SS). To smooth the cepstrum even further, another denoising method is used, namely wavelet denoising. However, this paper’s main intent is investigating the possibilities of cepstral editing and not giving an in-depth review about denoising theory, thus readers interested in spectral subtraction and wavelet denoising are referred to Refs.[19, 20] for more information. 2.4.3

Peak detection

After long-pass liftering and denoising, a fixed threshold of 3σ(/std) is calculated from the residual cepstrum cr (n) as follows: threshold = E[cr (n)] + 3 × std[cr (n)],

(10)

where E[.] and std[.] denote respectively the expectation operator and the standard deviation operator. Next a vector m is constructed, based on all the values greater than the threshold, containing all the corresponding sample indices: m = {∀n|cr (n) > threshold} 2.4.4

(11)

Cepstrum liftering

The actual liftering operation in the cepstrum is based on a simple notch lifter. The notch lifter lnotch (n) is generated based on the vector m which contains all the locations to be liftered. It should be noted though that the notch width is an important parameter that has a large influence on the performance of the cepstrum editing procedure. It shouldn’t be chosen too small in order to account for possible speed variations but it shouldn’t be too large either since that could lead to overly distorting the signal content. Until now, no strict guideline has been found regarding this issue and for the time being it is solved by visual inspection of the resulting spectra.

3.

Performance ACEP

In order to properly assess the efficiency of the cepstrum editing procedure, it is tested on simulated signals which try to give a qualitative representation of real world measurements. A simulated model is made in the assumption that the main use of ACEP is to pre-whiten the signal for easier bearing fault detection in a deterministic environment. A representation of the used model can be seen on Fig.3. The model takes into account the contribution of the noise-free bearing signal xbearing and the deterministic excitation sources xdeterministic . Next, the system responses ybearing and ydeterministic ICSV23, Athens (Greece), 10-14 July 2016

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are found by convolving the sources with the transfer path impulse responses of the system. Finally, the model output is the measured signal ymeasured which consists of the system responses ybearing and ydeterministic combined with additive white Gaussian noise enoise which represents the measurement noise.

Figure 3: Schematic diagram of the used simulation model. To account for the simulation of a realistic bearing fault signal, the model makes use of research conducted by Antoni and Randall [3]. In their paper they suggest modelling the vibration signal of a faulty rolling element bearing as a series of impulse responses of a single-degree-of-freedom (SDOF) system with random jitter on the fundamental period and random amplitude modulation. Additionally the model takes into account the system resonances excited by the faulty bearing, and the damping ratio. The following model is synthesized as a SDOF system with a resonance frequency of 1500 Hz and a damping ratio of 8%. The Dirac impulse period corresponds to a frequency of 32 Hz with 5% random jitter on the period and 10% random amplitude modulation. The resulting simulated signal has a duration of two seconds and can be seen on Fig.4(a) without noise and on Fig.4(b) with added white Gaussian noise of SNR=-10dB. Figure 4(c) shows the log amplitude spectrum of the noisy bearing signal with a clear increase in amplitude around the resonance. On the right side, figure.4(d) displays the deterministic time signal added to the system model. The added deterministic content consists of five harmonic families with fundamental frequencies of 22 Hz, 37 Hz, 50 Hz, 60 Hz and 81 Hz. Each harmonic has random amplitude and random phase. Finally, figure 4(e) and (f) show respectively the resulting time signal and the corresponding log amplitude spectrum which displays the dominating deterministic content together with the much weaker bearing content.

Figure 4: (a) Simulated rolling element bearing fault signal at 32 Hz with 5% random jitter, 10% random amplitude modulation and a resonance of 1500 Hz. (b) Same signal with additive white Gaussian noise (SNR=-10 dB). (c) Log amplitude spectrum of noisy bearing signal. (d) Deterministic time signal consisting of harmonics of 22, 37, 50, 60 and 81 Hz. (e) Resulting time signal of all three components, bearing fault, noise and harmonics. (f) Log amplitude spectrum of the resulting signal. To better comprehend the internal working of the ACEP method, figure 5 displays some crucial 6

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steps of the method. In Fig.5(a) the cepstrum of the unedited synthesized signal can be seen before ACEP. While the peaks are relatively distinguishable for this simulated model from the cepstral noise, it can be seen that for the long-pass liftered and denoised cepstrum in Fig.5(b) the thresholding is more straightforward, making this a recommended step for both simulated as measured data. Additionally the cepstral peaks corresponding to the harmonic families can be clearly discerned in Fig.5(b). Based on the detected peaks in the denoised cepstrum, a notch lifter is generated and can be observed in Fig.5(c). Next, the real cepstrum is liftered and transformed back to the time and spectral domain. The results of this operation can be seen on Fig.5(d) and (e) in respectively the time domain and spectral domain. The latter figure clearly exposes the bearing resonance when compared to Fig.4(f) because of a very strong reduction in harmonic amplitudes. To compare the effectiveness of the ACEP method, the squared envelope amplitude spectra before and after ACEP are displayed in Fig.5(f). It can be observed that the edited signal reveals the presence of the fundamental bearing fault frequency of 32 Hz and its harmonics, while the original unedited signal doesn’t produce any meaningful results.

Figure 5: (a) Real cepstrum of signal before editing (zoomed). (b) Cepstrum after long-pass liftering and denoising (zoomed). (c) Generated notch lifter based on the long-pass liftered, denoised cepstrum (zoomed). (d) Time signal after ACEP. (e) Spectrum after ACEP. (f) Comparison of squared envelope spectra before and after ACEP (zoomed).

4.

Conclusion

This paper investigates the performance of automated cepstrum editing in separating periodic and random components, more in particular deterministic content like harmonics and cyclostationary signals like typical bearing fault signals. The cepstrum has the interesting property of collecting all the periodic components in the log amplitude spectrum into so-called cepstral peaks. These peaks can then be edited to remove the corresponding harmonics. The cepstrum is also flexible for editing purposes in the sense that only certain periodic families can be chosen to be removed. Analysis of the automated cepstrum procedure on a simulated bearing fault model corrupted with dominant harmonics and white Gaussian noise proves that the ACEP method performs very well in reducing the discrete content and thus revealing the underlying bearing signal through the use of envelope analysis. Further research will focus on assessing the performance of ACEP on experimental measurement setups. ICSV23, Athens (Greece), 10-14 July 2016

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REFERENCES 1. Graney, B. P. and Starry, K. Rolling element bearing analysis, Materials Evaluation, 70 (1), 78, (2012). 2. Technologies, A. I. Lubricant failure = bearing failure, Machinery Lubrication Magazine, (1), (2009). 3. Antoni, J. and Randall, R. A stochastic model for simulation and diagnostics of rolling element bearings with localized faults, Journal of vibration and acoustics, 125 (3), 282–289, (2003). 4. McFadden, P. and Toozhy, M. Application of synchronous averaging to vibration monitoring of rolling element bearings, Mechanical Systems and Signal Processing, 14 (6), 891–906, (2000). 5. Antoni, J. and Randall, R. Unsupervised noise cancellation for vibration signals: part i evaluation of adaptive algorithms, Mechanical Systems and Signal Processing, 18 (1), 89–101, (2004). 6. Sawalhi, N. and Randall, R. B. The application of spectral kurtosis to bearing diagnostics, Proceedings of ACOUSTICS, pp. 3–5, (2004). 7. Antoni, J. and Randall, R. Unsupervised noise cancellation for vibration signals: part ii a novel frequency domain algorithm, Mechanical Systems and Signal Processing, 18 (1), 103–117, (2004). 8. Randall, R. B. and Sawalhi, N., (2014), Cepstral removal of periodic spectral components from time signals. Advances in Condition Monitoring of Machinery in Non-Stationary Operations, pp. 313–324, Springer. 9. Ompusunggu, A. P. Automated cepstral editing procedure (acep) as a signal pre-processing in vibrationbased bearing fault diagnostics. 10. Kilundu, B., Ompusunggu, A. P., Elasha, F. and Mba, D. Effect of parameters setting on performance of discrete component removal (dcr) methods for bearing faults detection, Proceedings of the European conference of the Prognostics and Health Management (PHM) Society, Nantes (France), 8th-10th July, (2014). 11. Bogert, B. P., Healy, M. J. and Tukey, J. W. The quefrency alanysis of time series for echoes: Cepstrum, pseudo-autocovariance, cross-cepstrum and saphe cracking, Proceedings of the symposium on time series analysis, vol. 15, pp. 209–243, chapter, (1963). 12. Oppenheim, A. V., Schafer, R. W. and Stockham Jr, T. G. Nonlinear filtering of multiplied and convolved signals, Audio and Electroacoustics, IEEE Transactions on, 16 (3), 437–466, (1968). 13. Childers, D. G., Skinner, D. P. and Kemerait, R. C. The cepstrum: A guide to processing, Proceedings of the IEEE, 65 (10), 1428–1443, (1977). 14. Randall, R. and Hee, J. Cepstrum analysis, Wireless World, 88, 77–80, (1982). 15. Polydoros, A. and Fam, A. T. The differential cepstrum: definition and properties, Proc. IEEE Int. Symp. Circuits Syst, pp. 77–80, (1981). 16. Antoni, J., Daniere, J. and Guillet, F. Blind identification of nonminimum phase systems using the mean differential cepstrum, Signal Processing Conference, 2000 10th European, pp. 1–4, IEEE, (2000). 17. Randall, R. B. A history of cepstrum analysis and its application to mechanical problems, International Conference at Institute of Technology of Chartres, France, pp. 11–16, (2013). 18. Randall, R. B. and Sawalhi, N. A new method for separating discrete components from a signal, Sound and Vibration, 45 (5), 6, (2011). 19. Boll, S. F. Suppression of acoustic noise in speech using spectral subtraction, Acoustics, Speech and Signal Processing, IEEE Transactions on, 27 (2), 113–120, (1979). 20. Pasti, L., Walczak, B., Massart, D. and Reschiglian, P. Optimization of signal denoising in discrete wavelet transform, Chemometrics and intelligent laboratory systems, 48 (1), 21–34, (1999).

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ICSV23, Athens (Greece), 10-14 July 2016